The approximate Loebl-Koml\'os-S\'os Conjecture I: The sparse decomposition

TL;DR

This paper introduces a new decomposition technique for sparse graphs and proves a relaxed version of the Loebl-Komlos-Sos Conjecture, showing that certain degree conditions guarantee the embedding of any tree of a given size.

Contribution

It presents a novel decomposition method for sparse graphs and establishes a relaxed embedding condition for trees, advancing the understanding of graph embedding problems.

Findings

Decomposition of graphs into high-degree vertices, regular pairs, and expansion objects.

Proof of a relaxed condition ensuring tree embedding in graphs with certain degree properties.

Introduction of a new technique applicable to sparse graph analysis.

Abstract

In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemer\'edi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse graphs $G$, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree.